# Institute of Mathematics

## On the regularity and defect sequence of monomial and binomial ideals

#### Keivan Borna, Abolfazl Mohajer

###### Received October 4, 2017.   Published online October 16, 2018.

Abstract:  When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$, with homogeneous maximal ideal $\mathfrak{m}$, it is known that for an ideal $I$ of $S$, the regularity of powers of $I$ becomes eventually a linear function, i.e., ${\rm reg}(I^m)=dm+e$ for $m\gg0$ and some integers $d$, $e$. This motivates writing ${\rm reg}(I^m)=dm+e_m$ for every $m\geq0$. The sequence $e_m$, called the \emph{defect sequence} of the ideal $I$, is the subject of much research and its nature is still widely unexplored. We know that $e_m$ is eventually constant. In this article, after proving various results about the regularity of monomial ideals and their powers, we give several bounds and restrictions on $e_m$ and its first differences when $I$ is a primary monomial ideal. Our theorems extend the previous results about $\mathfrak{m}$-primary ideals in the monomial case. We also use our results to obtatin information about the regularity of powers of a monomial ideal using its primary decomposition. Finally, we study another interesting phenomenon related to the defect sequence, namely that of regularity jump, where we give an infinite family of ideals with regularity jumps at the second power.
Keywords:  Castelnuovo-Mumford regularity; powers of ideal; defect sequence
Classification MSC:  13D02, 13P10
DOI:  10.21136/CMJ.2018.0458-17

References:
[1] D. Berlekamp: Regularity defect stabilization of powers of an ideal. Math. Res. Lett. 19 (2012), 109-119. DOI 10.4310/MRL.2012.v19.n1.a9 | MR 2923179 | Zbl 1282.13026
[2] K. Borna: On linear resolution of powers of an ideal. Osaka J. Math. 46 (2009), 1047-1058. MR 2604920 | Zbl 1183.13016
[3] M. P. Brodmann, R. Y. Sharp: Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511629204 | MR 1613627 | Zbl 0903.13006
[4] M. Chardin: Some results and questions on Castelnuovo-Mumford regularity. Syzygies and Hilbert Functions (I. Peeva, ed.). Lecture Notes in Pure and Applied Mathematics 254, Chapman & Hall/CRC, Boca Raton, 2007, 1-40. MR 2309925 | Zbl 1127.13014
[5] A. Conca: Regularity jumps for powers of ideals. Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects (A. Corso et al., eds.). Lecture Notes in Pure and Applied Mathematics 244, Chapman & Hall/CRC, Boca Raton, 2006, 21-32. MR 2184787 | Zbl 1080.13015
[6] S. D. Cutkosky, J. Herzog, N. V. Trung: Asymptotic behaviour of the Castelnuovo-Mumford regularity. Compos. Math. 118 (1999), 243-261. DOI 10.1023/A:1001559912258 | MR 1711319 | Zbl 0974.13015
[7] H. T. Hà, N. V. Trung, T. N. Trung: Depth and regularity of powers of sums of ideals. Math. Z. 282 (2016), 819-838. DOI 10.1007/s00209-015-1566-9 | MR 3473645 | Zbl 1345.13006
[8] V. Kodiyalam: Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Am. Math. Soc. 128 (2000), 407-411. DOI 10.1090/S0002-9939-99-05020-0 | MR 1621961 | Zbl 0929.13004

Affiliations:   Keivan Borna, Faculty of Mathematics and Computer Science, Kharazmi University, No. 43. South Mofatteh Ave., Tehran, 15719-14911, Iran, e-mail: borna@khu.ac.ir; Abolfazl Mohajer, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, D 55128 Mainz, Germany, e-mail: mojaher@uni-mainz.de

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